Let G = Z4 + U(4), H = <(2,3)> and K = <(2,1)>. Show that G/H is not isomorphic to G/K.
Hint: note that both G/H and G/K have 4 elements. show that every non-identity element of G/K has order 2 and find an element of G/H which has order 4.
this is not a part of solution but the above also proves that $\displaystyle G/K \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$ and $\displaystyle G/H \cong \mathbb{Z}_4.$
Do you have LaGrange's Thorem yet?
$\displaystyle |G/H|=\frac{|G|}{|H|}$
As you noted $\displaystyle |G|=|\mathbb{Z}_4 \oplus U_4|=8$
Direct computation shows $\displaystyle |H|=|<2,3>|=|\{(2,3) , (0,1) \}|= 2$
Similarly,
$\displaystyle |K|=|<2,1>|=|\{(2,1) , (0,1) \}|= 2$
So you see by the above,
$\displaystyle |G/H|=\frac{|G|}{|H|}=\frac{8}{2}=4$
$\displaystyle |G/H|=\frac{|G|}{|K|}=\frac{8}{2}=4$
Follow NCA's hints to see why $\displaystyle H \not \cong K$